Camera motion is dependent on a few parameters. First of all, the exposure speed. The longer the shutter is open, the more likely that movement will be noticed. The second is the focal length of the camera. The longer the lens is, the more noticeable the movement is. A rule of thumb for amateur photographers shooting 35 mm film is never to exceed the exposure time beyond the focal length, so that for a 30 mm lens, not to shoot slower than 1/30th of a second. The third criteria is the subject itself. Flat areas, or low frequency data, is less likely to be degraded as much as high frequency data.
Historically, the problem was addressed by anchoring the camera, such as with the use of a tripod or monopod, or stabilizing it such as with the use of gyroscopic stabilizers in the lens or camera body, or movement of the sensor plane to counteract the camera movement.
Mathematically, the motion blurring can be explained as applying a Point Spread Function, or PSF, to each point in the object. This PSF represent the path of the camera, during the exposure integration time. Motion PSF is a function of the motion path and the motion speed, which determines the integration time, or the accumulated energy for each point.
A hypothetical example of such a PSF is illustrated in FIG. 3-a and 3-b. FIG. 3-b is a projection of FIG. 3-a. In FIGS. 3-a and 3-b, the PSF is depicted by 410 and 442 respectively. The pixel displacement in x and y directions are depicted by blocks 420 and 421 respectively for the X axis and 430 and 432 for the Y axis respectively. The energy 440 is the third dimension of FIG. 3-a. Note that the energy is the inverse of the differential speed in each point, or directly proportional to the time in each point. In other words, the longer the camera is stationary at a given location, the longer the integration time is, and thus the higher the energy packed. This may also be depicted as the width of the curve 442 in a X-Y projection.
Visually, when referring to images, in a simplified manner, FIG. 3-c illustrates what would happen to a pinpoint white point in an image blurred by the PSF of the aforementioned Figures. In a picture, such point of light surrounded by black background will result in an image similar to the one of FIG. 3-c. In such image, the regions that the camera was stationary longer, such as 444 will be brighter than the region where the camera was stationary only a fraction of that time. Thus such image may provide a visual speedometer, or visual accelerometer. Moreover, in a synthetic photographic environment such knowledge of a single point, also referred to as a delta-function could define the PSF.
Given:                a two dimensional image I represented by I(x,y)        a motion point spread function MPSF(I)        The degraded image I′(x,y) can be mathematically defined as the convolution of I(X,Y) and MPSF(x,y) orI′(x,y)=I(x,y)(MPSF(x,y)  (Eq. 1)        
or in the integral form for a continuous functionI(x,y)=∫∫(I(x−x′,y−y′)MPSF(x′y′)∂x′∂y′  (Eq. 2)
and for a discrete function such as digitized images:
                                          I            ′                    ⁡                      (                          m              ,              n                        )                          =                              ∑            j                    ⁢                                    ∑              k                        ⁢                                          I                ⁡                                  (                                                            m                      -                      j                                        ,                                          n                      -                      k                                                        )                                            ⁢                              MPSF                ⁡                                  (                                      j                    ,                    k                                    )                                                                                        (                  Eq          .                                          ⁢          3                )            
Another well known PSF in photography and in optics in general is blurring created by de-focusing. The different is that de-focusing can usually be depicted by a symmetrical Gaussian shift invariant PSF, while motion de-blurring is not.
The reason why motion de-blurring is not shift invariant is that the image may not only shift but also rotate. Therefore, a complete description of the motion blurring is an Affine transform that combines shift and rotation based on the following transformation:
                              [                                                    u                                                                    v                                                                    1                                              ]                =                  [                                                                      Cos                  ⁢                                                                          ⁢                  ω                                                                              Sin                  ⁢                                                                          ⁢                  ω                                                                              Δ                  ⁢                                                                          ⁢                  x                                                                                                                          -                    Sin                                    ⁢                                                                          ⁢                  ω                                                                              cos                  ⁢                                                                          ⁢                  ω                                                                              Δ                  ⁢                                                                          ⁢                  y                                                                                    0                                            0                                            1                                              ]                                    (                  Eq          .                                          ⁢          4                )            
The PSF can be obtained empirically as part of a more generic field such as system identification. For linear systems, the PSF can be determined by obtaining the system's response to a known input and then solving the associated inversion problems.
The known input can be for an optical system, a point, also mathematically defined in the continuous world as a delta function δ(x), a line, an edge or a corner.
An example of a PSF can be found in many text books such as “Deconvolution of Images and Spectra” 2nd. Edition, Academic Press, 1997, edited by Jannson, Peter A. and “Digital Image Restoration”, Prentice Hall, 1977 authored by Andrews, H. C. and Hunt, B. R.
The process of de-blurring an image is done using de-convolution which is the mathematical form of separating between the convolve image and the convolution kernel. However, as discussed in many publications such as Chapter 1 of “Deconvolution of Images and Spectra” 2nd. Edition, Academic Press, 1997, edited by Jannson, Peter A., the problem of de-convolution can be either unsolvable, ill-posed or ill-conditioned. Moreover, for a physical real life system, an attempt to find a solution may also be exacerbated in the presence of noise or sampling.
One may mathematically try and perform the restoration via de-convolution means without the knowledge of the kernel or in this case the PSF. Such methods known also as blind de-convolution. The results of such process with no a-priori knowledge of the PSF for a general optical system are far from acceptable and require extensive computation. Solutions based on blind de-convolution may be found for specific circumstances as described in “Automatic multidimensional deconvolution” J. Opt. Soc. Am. A, vol. 4(1), pp. 180-188, January 1987 to Lane et al, “Some Implications of Zero Sheets for Blind Deconvolution and Phase Retrieval”, J. Optical Soc. Am. A, vol. 7, pp. 468-479, 1990 to Bates et al, Iterative blind deconvolution algorithm applied to phase retrieval”, J. Opt. Soc. Am. A, vol. 7(3), pp. 428-433, March 1990 to Seldin et al and “Deconvolution and Phase Retrieval With Use of Zero Sheets,” J. Optical Soc. Am. A, vol. 12, pp. 1,842-1,857, 1995 to Bones et al. However, as known to those familiar in the art of image restoration, and as explained in “Digital Image Restoration”, Prentice Hall, 1977 authored by Andrews, H. C. and Hunt, B. R., blurred images can be substantially better restored when the blur function is known.
The article “Motion Deblurring Using Hybrid Imaging”, by Moshe Ben-Ezra and Shree K. Nayar, from the Proceedings IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2003, determines the PSF of a blurred image by using a hybrid camera which takes a number of relatively sharp reference images during the exposure period of the main image. However, this requires a special construction of camera and also requires simultaneous capture of images. Thus this technique is not readily transferable to cheap, mass-market digital cameras.
It is an object of the invention to provide an improved technique for determining a camera motion blur function in a captured digital image which can take advantage of existing camera functionality and does not therefore require special measurement hardware (although the use of the invention in special or non-standard cameras is not ruled out).